Bit mapping scheme for an ldpc coded 32apsk system

ABSTRACT

A digital communication system, having a transmitter to transmit a digital signal; and a receiver to receive the digital signal; wherein the digital signal utilizes a 32APSK system with FEC coding, and the signal is bit-mapped using gray mapping, and bits of the digital signal are ordered based on the values of a log likelihood ratio from a communications channel.

RELATED APPLICATIONS

This application relates to application Ser. No. ______, titled “A BitMapping Scheme for an LDPC Coded 16APSK System” filed on ______, andapplication Ser. No. ______ titled “An Interleaving Scheme for an LDPCCoded 32APSK System” filed on ______.

FIELD OF THE INVENTION

The invention relates to digital communications and in particular to abit mapping scheme for an LDPC coded 32APSK System.

BACKGROUND OF THE INVENTION

Forward Error Control (FEC) coding is used by communications systems toensure reliable transmission of data across noisy communicationchannels. Based on Shannon's theory, these communication channelsexhibit a fixed capacity that can be expressed in terms of bits persymbol at a given Signal to Noise Ratio (SNR), which is defined as theShannon limit. One of the research areas in communication and codingtheory involves devising coding schemes offering performance approachingthe Shannon limit while maintaining a reasonable complexity. It has beenshown that LDPC codes using Belief Propagation (BP) decoding provideperformance close to the Shannon limit with tractable encoding anddecoding complexity.

In a recent paper Yan Li and William Ryan, “Bit-Reliability Mapping inLDPC-Codes Modulation systems”, IEEE Communications Letters, vol. 9, no.1, January 2005, the authors studied the performance of LDPC-codedmodulation systems with 8PSK. With the authors' proposed bit reliabilitymapping strategy, about 0.15 dB performance improvement over thenon-interleaving scheme is achieved. Also the authors show that graymapping is more suitable for high order modulation than other mappingscheme such as natural mapping.

BRIEF SUMMARY OF THE INVENTION

Various embodiments of the present invention are directed to a bitmapping scheme in a 32APSK modulation system. The techniques of theseembodiments are particularly well suited for use with LDPC codes.

LDPC codes were first described by Gallager in the 1960s. LDPC codesperform remarkably close to the Shannon limit. A binary (N, K) LDPCcode, with a code length N and dimension K, is defined by a parity checkmatrix H of (N-K) rows and N columns. Most entries of the matrix H arezeros and only a small number the entries are ones, hence the matrix His sparse. Each row of the matrix H represents a check sum, and eachcolumn represents a variable, e.g., a bit or symbol. The LDPC codesdescribed by Gallager are regular, i.e., the parity check matrix H hasconstant-weight rows and columns.

Regular LDPC codes can be extended to form irregular LDPC codes, inwhich the weight of rows and columns vary. An irregular LDPC code isspecified by degree distribution polynomials v(x) and c(x), which definethe variable and check node degree distributions, respectively. Morespecifically, the irregular LDPC codes may be defined as follows:

$\begin{matrix}{{{v(x)} = {\sum\limits_{j = 1}^{d_{v\; \max}}{v_{j}x^{j - 1}}}},{and}} & (1) \\{{{c(x)} = {\sum\limits_{j = 1}^{d_{c\; \max}}{c_{j}x^{j - 1}}}},} & (2)\end{matrix}$

where the variables d_(ν max) and d_(c max) are a maximum variable nodedegree and a check node degree, respectively, and ν_(j)(c_(j))represents the fraction of edges emanating from variable (check) nodesof degree j. While irregular LDPC codes can be more complicated torepresent and/or implement than regular LDPC codes, it has been shown,both theoretically and empirically, that irregular LDPC codes withproperly selected degree distributions outperform regular LDPC codes.FIG. 1 illustrates a parity check matrix representation of an exemplaryirregular LDPC code of codeword length six.

LDPC codes can also be represented by bipartite graphs, or Tannergraphs. In Tanner graph, one set of nodes called variable nodes (or bitnodes) corresponds to the bits of the codeword and the other set ofnodes called constraints nodes (or check nodes) corresponds the set ofparity check constrains which define the LDPC code. Bit nodes and checknodes are connected by edges, and a bit node and a check node are saidto be neighbors or adjacent if they are connected by an edge. Generally,it is assumed that a pair of nodes is connected by at most one edge.

FIG. 2 illustrates a bipartite graph representation of the irregularLDPC code illustrated in FIG. 1.

LDPC codes can be decoded in various ways such as majority-logicdecoding and iterative decoding. Because of the structures of theirparity check matrices, LDPC codes are majority-logic decodable. Althoughmajority-logic decoding requires the least complexity and achievesreasonably good error performance for decoding some types of LDPC codeswith relatively high column weights in their parity check matrices(e.g., Euclidean geometry LDPC and projective geometry LDPC codes),iterative decoding methods have received more attention due to theirbetter performance versus complexity tradeoffs. Unlike majority-logicdecoding, iterative decoding processes the received symbols recursivelyto improve the reliability of each symbol based on constraints thatspecify the code. In a first iteration, an iterative decoder only uses achannel output as input, and generates reliability output for eachsymbol.

Subsequently, the output reliability measures of the decoded symbols atthe end of each decoding iteration are used as inputs for the nextiteration. The decoding process continues until a stopping condition issatisfied, after which final decisions are made based on the outputreliability measures of the decoded symbols from the last iteration.According to the different properties of reliability measures usedduring each iteration, iterative decoding algorithms can be furtherdivided into hard decision, soft decision and hybrid decisionalgorithms. The corresponding popular algorithms are iterativebit-flipping (BF), belief propagation (BP), and weighted bit-flipping(WBF) decoding, respectively. Since BP algorithms have been proven toprovide maximum likelihood decoding when the underlying Tanner graph isacyclic, they have become the most popular decoding methods.

BP for LDPC codes is a type of message passing decoding. Messagestransmitted along the edges of a graph are log-likelihood ratio

$({LLR})^{\log \frac{p_{0}}{p_{1}}}$

associated with variable nodes corresponding to codeword bits. In thisexpression p₀ and p₁ denote the probability that the associated bitvalue becomes either a 0 or a 1, respectively. BP decoding generallyincludes two steps, a horizontal step and a vertical step. In thehorizontal step, each check node c_(m) sends to each adjacent bit b_(n)a check-to-bit message which is calculated based on all bit-to-checkmessages incoming to the check c_(m) except one from bit b_(n). In thevertical step, each bit node b_(n) sends to each adjacent check nodec_(m) a bit-to-check message which is calculated based on allcheck-to-bit messages incoming to the bit b_(n) except one from checknode c_(m). These two steps are repeated until a valid codeword is foundor the maximum number of iterations is reached.

Because of its remarkable performance with BP decoding, irregular LDPCcodes are among the best for many applications. Various irregular LDPCcodes have been accepted or being considered for various communicationand storage standards, such as DVB-S2/DAB, wireline ADSL, IEEE 802.11n,and IEEE 802.16.

The threshold of an LDPC code is defined as the smallest SNR value atwhich, as the codeword length tends to infinity, the bit errorprobability can be made arbitrarily small. The value of threshold of anLDPC code can be determined by analytical tool called density evolution.

The concept of density evolution can also be traced back to Gallager'sresults. To determine the performance of BF decoding, Gallager derivedformulas to calculate the output BER for each iteration as a function ofthe input BER at the beginning of the iteration, and then iterativelycalculated the BER at a given iteration. For a continuous alphabet, thecalculation is more complex. The probability density functions (pdf's)of the belief messages exchanged between bit and check nodes need to becalculated from one iteration to the next, and the average BER for eachiteration can be derived based on these pdf's. In both check nodeprocessing and bit node processing, each outgoing belief message is afunction of incoming belief messages. For a check node of degree d_(c),each outgoing message U can be expressed by a function of d_(c)−1incoming messages,

U=F _(c)(V ₁ , V ₂ , . . . , V _(d) _(c) ⁻¹),

where F_(c) denotes the check node processing function which isdetermined from BP decoding. Similarly, for bit node of degree d_(ν),each outgoing message V can be expressed by a function of d_(ν)−1incoming coming messages and the channel belief message U_(ch),

V=F _(V)(U _(ch) , U ₁ , U ₂ , . . . , U _(d) _(ν) ⁻¹),

where F_(ν) denotes the bit node processing function. Although for bothcheck and bit node processing, the pdf of an outgoing message can bederived based on the pdf's of incoming messages for a given decodingalgorithm, there may exist an exponentially large number of possibleformats of incoming messages. Therefore the process of density evolutionseems intractable. Fortunately, it has been proven in that for a givenmessage-passing algorithm and noisy channel, if some symmetry conditionsare satisfied, then the decoding BER is independent of the transmittedsequence x. That is to say, with the symmetry assumptions, the decodingBER of all-zero transmitted sequence x=1 is equal to that of anyrandomly chosen sequence, thus the derivation of density evolution canbe considerably simplified. The symmetry conditions required byefficient density evolution are channel symmetry, check node symmetry,and bit node symmetry. Another assumption for the density evolution isthat the Tanner graph is cyclic free.

According to these assumptions, the incoming messages to bit and checknodes are independent, and thus the derivation for the pdf of theoutgoing messages can be considerably simplified. For many LDPC codeswith practical interests, the corresponding Tanner graph containscycles. When the minimum length of a cycle (or girth) in a Tanner graphof an LDPC code is equal to 4×l, then the independence assumption doesnot hold after the l-th decoding iteration with the standard BPdecoding. However, for a given iteration number, as the code lengthincreases, the independence condition is satisfied for an increasingiteration number. Therefore, the density evolution predicts theasymptotic performance of an ensemble of LDPC codes and the “asymptotic”nature is in the sense of code length.

A bit mapping scheme is provided for low density parity check (LDPC)encoded bits in 32APSK modulation systems. The disclosed bit mappingscheme provides good threshold of LDPC codes. Furthermore the bitmapping scheme can facilitate design of interleaving arrangement in32APSK modulation system.

To propose a bit mapping approach for LDPC coded 32APSK systems. Thedisclosed bit mapping offers good performance of LDPC coded 32APSKsystem and simplifies interleaving arrangement in 32APSK systems.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the corresponding drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 is a parity check matrix representation of an exemplary irregularLDPC code of codeword length six.

FIG. 2 illustrates a bipartite graph representation of the irregularLDPC code illustrated in FIG. 1.

FIG. 3 illustrates the bit mapping block in 32APSK modulation, accordingto various embodiments of the invention.

FIG. 4 illustrates a bit map for 32APSK symbol, according to variousembodiments of the invention.

FIG. 5 depicts an exemplary communications system which employs LDPCcodes and 32APSK modulation, according to various embodiments of theinvention.

FIG. 6 depicts an exemplary transmitter employing 32APSK modulation inFIG. 5, according to various embodiments of the invention.

FIG. 7 depicts an exemplary receiver employing 32APSK demodulation inFIG. 5, according to various embodiments of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Referring to the accompanying drawings, a detailed description will begiven of exemplary encoded bit mapping methods using LDPC codesaccording to various embodiments of the invention.

Although the invention is described with respect to LDPC codes, it isrecognized that the bit mapping approach can be utilized with othercodes. Furthermore, it is recognized that this approach can beimplemented with uncoded systems.

FIG. 5 is a diagram of a communications system employing LDPC codes with32APSK modulation, according to an embodiment of the present invention.A communications system includes a transmitter 501 which generatessignal waveforms across a communication channel 502 to a receiver 503.The transmitter 501 contains a message source producing a discrete setof possible messages. Each of these messages corresponds a signalwaveform. The waveforms enter the channel 502 and are corrupted bynoise. LDPC codes are employed to reduce the disturbances introduced bythe channel 502 and a 32APSK modulation scheme is employed to transformLDPC encoded bits to signal waveforms.

FIG. 6 depicts an exemplary transmitter in the communications system ofFIG. 5 which employs LDPC codes and 32APSK modulation. The LDPC encoder602 encodes information bits from source 601 into LDPC codewords. Themapping from each information block to each LDPC codeword is specifiedby the parity check matrix (or equivalently the generator matrix) of theLDPC code. The LDPC codeword is interleaved and modulated to signalwaveforms by the interleaver/modulator 603 based on a 32APSK bit mappingscheme. These signal waveforms are sent to a transmit antenna 604 andpropagated to a receiver shown in FIG. 7.

FIG. 7 depicts an exemplary receiver in FIG. 5 which employs LDPC codesand 32APSK demodulator. Signal waveforms are received by the receivingantenna 701 and distributed to deznodulator/deinterleavor 702. Signalwaveforms are demodulated by demodulator and deinterleaved bydeinterleavor and then distributed to a LDPC decoder 703 whichiteratively decodes the received messages and output estimations of thetransmitted codeword. The 32APSK demodulation rule employed by thedemodulator/deinterleaver 702 should match with the 32APSK modulationrule employed by the interleaver/modulator 603.

According to various embodiments of the invention, as shown in FIG. 3,the 32APSK bit-to-symbol mapping circuit may utilize five bits (b5 i, b5i+1, b5 i+2, b5 i+3, b5 i+4) each iteration and map them into an I valueand a Q value, with i=0, 1, 2, . . . . The bit mapping logic is shown inFIG. 4. The bit mapping according to various embodiments of theinvention is defined as follows:

$\left( {{I(i)},{Q(i)}} \right) = \left( {\begin{matrix}{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,1} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,0} \right)} \\{\left( {R_{3},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,1} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,1} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,1} \right)}\end{matrix}\mspace{76mu} \left\{ {\begin{matrix}{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,1} \right)} \\{\left( {0,{- R_{3}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,0} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,1} \right)}\end{matrix}\mspace{135mu} \left( \begin{matrix}{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,1} \right)} \\{\left( {0,R_{3}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,0} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,0} \right)} \\{\left( {{- R_{3}},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,1} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,1} \right)}\end{matrix} \right.} \right.} \right.$

According to various embodiments of the invention, the bit mappingscheme of FIG. 4 may use gray mapping which means the binaryrepresentations of adjacent points differ by only one bit. Densityevolution analysis shows that given an LDPC coded 32APSK system, graymapping scheme can provide the best threshold. The bit mapping scheme ofFIG. 4 also arranges bit in an order based on the values of loglikelihood ratio from communications channel. This arrangementsimplifies the design of interleaving scheme for 32APSK system.

Although the invention has been described by the way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

1. A method of bit mapping in a 32APSK system, the method comprising:transmitting a digital signal from a transmitter; and receiving thedigital signal at a receiver; wherein the digital signal utilizes a32APSK system, and the signal is bit-mapped prior to the transmittingaccording to the following formula:$\left( {{I(i)},{Q(i)}} \right) = \left( {\begin{matrix}{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,1} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,0} \right)} \\{\left( {R_{3},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,1} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,1} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,1} \right)}\end{matrix}\mspace{76mu} \left\{ {\begin{matrix}{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,1} \right)} \\{\left( {0,{- R_{3}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,0} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,1} \right)}\end{matrix}\mspace{76mu} \left( \begin{matrix}{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,1} \right)} \\{\left( {0,R_{3}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,0} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,0} \right)} \\{\left( {{- R_{3}},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,1} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,1} \right)}\end{matrix} \right.} \right.} \right.$ wherein R1 is a radius of aninner ring, R2 is a radius of an intermediate ring and R3 is a radius ofan outer ring.
 2. The method of claim 1, wherein an FEC code is used. 3.A digital communication system, comprising: a transmitter to transmit adigital signal; wherein the digital signal utilizes a 32APSK system withFEC coding, and the signal is bit-mapped using gray mapping, and bits ofthe digital signal are ordered based on the values of a log likelihoodratio from a communications channel.
 4. The method of claim 3, whereinthe FEC code is regular LDPC code.
 5. The method of claim 3, wherein theFEC code is irregular LDPC code.
 6. The method of claim 3, wherein theFEC code is regular repeat-accumulate code.
 7. The method of claim 3wherein the FEC code is irregular repeat-accumulate code.
 8. A digitalcommunications system, comprising: a transmitter to modulate at leastone mapping group having five bits (b5 i, b5 i+1, b5 i+2, b5 i+3, b5i+4), for i=0, 1, 2, . . . , to 32APSK symbols based on formula:$\left( {{I(i)},{Q(i)}} \right) = \left( {\begin{matrix}{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,1} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,0} \right)} \\{\left( {R_{3},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,1} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,1} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,1} \right)}\end{matrix}\mspace{76mu} \left\{ {\begin{matrix}{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,1} \right)} \\{\left( {0,{- R_{3}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,0} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,1} \right)}\end{matrix}\mspace{76mu} \left( \begin{matrix}{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,1} \right)} \\{\left( {0,R_{3}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,0} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,0} \right)} \\{\left( {{- R_{3}},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,1} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,1} \right)}\end{matrix} \right.} \right.} \right.$ wherein R₁ is a radius of aninner ring, R₂ is a radius of an intermediate ring, and R₃ is a radiusof an outer ring.
 9. A digital communications system, comprising: areceiver to demodulate at least one mapped 32APSK symbol to anestimating message group having five bits (b5 i, b5 i+1, b5 i+2, b5 i+3,b5 i+4), for i=0, 1, 2, . . . , based on a 32APSK constellationspecification:$\left( {{I(i)},{Q(i)}} \right) = \left( {\begin{matrix}{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,1} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,0} \right)} \\{\left( {R_{3},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,1} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,1} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,1} \right)}\end{matrix}\mspace{76mu} \left\{ {\begin{matrix}{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,1} \right)} \\{\left( {0,{- R_{3}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,0} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,1} \right)}\end{matrix}\mspace{76mu} \left( \begin{matrix}{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,1} \right)} \\{\left( {0,R_{3}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,0} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,0} \right)} \\{\left( {{- R_{3}},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,1} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,1} \right)}\end{matrix} \right.} \right.} \right.$ wherein R₁ is a radius of aninner ring, R₂ is a radius of an intermediate ring, and R₃ is a radiusof an outer ring.
 10. A computer readable medium to store a computerprogram to map at least one group of five bits (b5 i, b5 i+1, b5 i+2, b5i+3, b5 i+4), for i=0, 1, 2, . . . , to a 32APSK symbol based onformula: $\left( {{I(i)},{Q(i)}} \right) = \left( {\begin{matrix}{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,0,1} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,0} \right)} \\{\left( {R_{3},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,0,1} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,0,1,1,1} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,0,1} \right)}\end{matrix}\mspace{76mu} \left\{ {\begin{matrix}{\left( {{R_{3}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,0} \right)} \\{\left( {{R_{3}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,0} \right)} \\{\left( {{R_{1}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,0,1} \right)} \\{\left( {0,{- R_{3}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,0} \right)} \\{\left( {{R_{3}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {0,1,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{R_{2}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{R_{3}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{R_{3}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,0,1,1} \right)}\end{matrix}\mspace{76mu} \left( \begin{matrix}{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{R_{1}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,0,1} \right)} \\{\left( {0,R_{3}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,0} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{R_{3}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/4} \right)}},{{- R_{2}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,0,1} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/8} \right)}},{{- R_{3}}{\sin \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,0} \right)} \\{\left( {{- R_{3}},0} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\cos \left( {\pi/4} \right)}},{{- R_{1}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,0,1} \right)} \\{\left( {{{- R_{3}}{\sin \left( {\pi/8} \right)}},{{- R_{3}}{\cos \left( {\pi/8} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,0} \right)} \\{\left( {{{- R_{3}}{\cos \left( {\pi/4} \right)}},{{- R_{3}}{\sin \left( {\pi/4} \right)}}} \right),} & {\left( {b_{5i},b_{{5i} + 1},b_{{5i} + 2},b_{{5i} + 3},b_{{5i} + 4}} \right) = \left( {1,1,1,1,1} \right)}\end{matrix} \right.} \right.} \right.$ wherein R₁ is a radius of aninner ring, R₂ is a radius of an intermediate ring, and R₃ is a radiusof an outer ring.